What is the equation of the parabola, whose vertex and focus are on the x-axis at distance a and b from the origin respectively ? `(bgtagt0)`

Options

A. `y^2=4(b-a)(x-a)`
B. `y^2=4(b-a)(x-b)`
C. `y^2=4(b-a)(y-a)`
D. `y^2=4(b-a)(y-b)`

Answer

To find the equation of the parabola whose vertex and focus are on the x-axis at distances \( a \) and \( b \) from the origin respectively, we can follow these steps:

### Step 1: Identify the coordinates of the vertex and focus
The vertex of the parabola is at the point \( (a, 0) \) and the focus is at the point \( (b, 0) \).

### Step 2: Understand the properties of the parabola
The distance between the vertex and the focus is denoted by \( \alpha \). For a parabola that opens to the right, the standard form of the equation is given by:
\[ y^2 = 4\alpha x \]
where \( \alpha \) is the distance from the vertex to the focus.

### Step 3: Calculate the value of \( \alpha \)
The distance \( \alpha \) can be calculated as:
\[ \alpha = b - a \]
This is because the focus \( (b, 0) \) is located to the right of the vertex \( (a, 0) \).

### Step 4: Shift the standard parabola equation
Since the vertex of our parabola is not at the origin but at \( (a, 0) \), we need to shift the standard equation \( y^2 = 4\alpha x \) to account for this. We replace \( x \) with \( (x - a) \):
\[ y^2 = 4\alpha (x - a) \]

### Step 5: Substitute \( \alpha \) into the equation
Now substituting \( \alpha = b - a \) into the equation gives:
\[ y^2 = 4(b - a)(x - a) \]

### Final Equation
Thus, the equation of the parabola whose vertex is at \( (a, 0) \) and focus at \( (b, 0) \) is:
\[ y^2 = 4(b - a)(x - a) \]